of GG-equivariant vector bundles over MM. The induced bundle construction gives a functor

L:Rep(H)→Vect(M,G)L: Rep(H) \to Vect(M,G)

But, if you think about it, you’ll notice there’s also a functor going back the other way:

R:Vect(M,G)→Rep(H)R: Vect(M,G) \to Rep(H)

If you give me a GG-equivariant vector bundle EE over MM, I can take its fiber over your favorite point xx, and I get a vector space — and this becomes a representation of the stabilizer group HH, thanks to how GG acts on EE.

This functor is simpler than the induced bundle construction!

Whenever we have functors going both ways between two categories, we should suspect that they’re adjoints. The simpler functor often amounts to ‘forgetting’ something. This forgetful functor is usually the right adjoint. It’s partner going the other way, the left adjoint, usually involves ‘constructing’ something instead of ‘forgetting’ something.

And indeed, that’s what’s happening here! Technically, this is to say that

hom(LV,F)≅hom(V,RF)hom(L V, F) \cong hom(V, R F)

Here VV is a representation of HH — note abuse of notation in calling it VV, which is the name for the vector space on which GG acts, instead of the more pedantic full name for a representation, which is something like s:G→GL(V)s: G \to GL(V).

Similarly, FF is a GG-equivariant vector bundle over MM — and this should be something like π:F→M\pi : F \to M, or something even more long-winded that gives a name to how GG acts on FF and MM.

LVL V is the induced bundle corresponding to VV.

RFR F is the fiber of FF over your favorite point xx, which becomes a representation of GG.

And this:

hom(LV,F)≅hom(V,RF)hom(L V, F) \cong hom(V, R F)

says that GG-equivariant vector bundle maps from LVL V to FF are in natural 1-1 correspondence with intertwining operators from VV to RFR F.

Now, whenever you see any sort of ‘forgetful’ process, you should wonder if it has a left adjoint, a construction which in some loose sense is the ‘reverse’ of forgetting. Why? Because these left adjoints tend to be important.

Endowed with this heuristic, as soon as you see there’s a rather obvious ‘forgetful’ process that takes a GG-equivariant vector bundle over MM and gives a representation of HH on the fiber over x∈Mx \in M, you will seek the ‘reverse’ process — and then you’ll rediscover the induced bundle construction!

And why is this so great? Well, there’s also a process that takes any representation of GG and restricts it to a representation of HH:

R′:Rep(G)→Rep(H)R': Rep(G) \to Rep(H)

And this too, has a left adjoint:

L′:Rep(H)→Rep(G)L' : Rep(H) \to Rep(G)

which is called the induced representation trick.

Detailed description

Given a groupGG with a subgroup HH, and a representationss of HH on a vector space VV, we define a left action of HH on the productG×VG\times V by h⋅(g,v)=(gh−1,s(h)v)h\cdot (g, v) = (g h^{-1}, s(h)v). We write [(g,v)][(g,v)] for the orbit, or equivalence class, that contains (g,v)(g,v).

We then define E=(G×V)/HE = (G\times V)/H as the set of orbits of that action of HH, M=G/HM = G/H as the set of left cosets of HH, and the projection π:E→M\pi: E\to M by π([(g,v)])=gH\pi ([(g,v)]) = g H, where of course it makes no difference if we re-describe the orbit [(g,v)][(g,v)] as [(gh−1,s(h)v][(g h^{-1}, s(h)v] for any h∈Hh\in H because (gh−1)H=gH(g h^{-1}) H = g H.

For each x∈Mx\in M, choose gg to be any element of GG such that x=gHx = g H. Define Ex=π−1(x)E_x = \pi^{-1}(x), and ϕg:V→Ex\phi_g:V\to E_x, ϕg(v)=[(g,v)]\phi_g(v) = [(g,v)].

The map ϕg\phi_g is one-to-one: if ϕg(v)=ϕg(w)\phi_g(v) = \phi_g(w), then [(g,v)]=[(g,w)][(g,v)]=[(g,w)], so for some h1∈Hh_1\in H, we have h1⋅(g,v)=(g,w)h_1\cdot (g,v) = (g,w), or (gh1−1,s(h1)v)=(g,w)(g h_1^{-1}, s(h_1)v) = (g,w); equating the first coordinates requires h1=eh_1=e, and ss is a representation so s(e)=1Vs(e)=1_V, and v=wv=w.

Since ϕg\phi_g is a bijection between ExE_x and the vector space VV, we can make ExE_x into a vector space by defining αp+βq≡ϕg(αϕg−1(p)+βϕg−1(q))\alpha p + \beta q \equiv \phi_g(\alpha \phi_g^{-1}(p) + \beta \phi_g^{-1}(q)), for all α,β∈ℝ,p,q∈Ex\alpha, \beta \in \mathbb{R}, p, q \in E_x. But is this independent of our choice of gg? If we chose ghg h instead of gg, we’d have ϕgh(v)=[(gh,v)]=[(g,s(h)v)]=ϕg(s(h)v)\phi_{g h}(v) = [(g h,v)] = [(g, s(h)v)] = \phi_g(s(h)v), so ϕgh=ϕg∘s(h)\phi_{g h}=\phi_g\circ s(h), and ϕgh−1=s(h−1)∘ϕg−1\phi_{g h}^{-1}=s(h^{-1})\circ \phi_g^{-1}. Then:

That is, π\pi is a GG-morphism. This also means that the action maps fibers to fibers, g1:E(gH)→Eg1⋅(gH)g_1:E_{(g H)}\to E_{g_1\cdot (g H)}. What’s more, the action of g1g_1 restricted to the fiber E(gH)E_{(g H)} is ϕg1g∘ϕg−1\phi_{g_1 g}\circ \phi_g^{-1}, passing from E(gH)→V→Eg1⋅(gH)E_{(g H)}\to V \to E_{g_1\cdot (g H)}, and this is linear simply by virtue of the way we’ve defined the vector space operations on the ExE_x.

We get a representation rr of GG on the vector space Γ(E)\Gamma(E) of sections of the bundle EE by:

(A genuine ∞-representation/∞-module over GG may be taken to be a an abelian ∞\infty-group object in Act(G)Act(G), but we can just as well work in the more general context of possibly non-linear representations, hence of actions.)

Properties

Unitarity

Beware! The chain of reasoning in this subsection is not complete, and I’m not confident that it’s entirely correct. I’m posting it half-finished in the hope that many hands will make lighter (and more accurate) work.

To be really thorough, we should verify that ⟨⟨⋅,⋅⟩⟩\lang \lang \cdot, \cdot \rang \rang is in fact an inner product, but this should follow directly from our definition of the vector space operations on ExE_x.

Now we need to show that the action of any g1∈Gg_1 \in G on the fiber E(gH)E_{(g H)} is unitary:

Finally, we need to define an inner product on Γ(E)\Gamma(E), and show that the representation rr is unitary. If we had a GG-invariant measure μ\mu on G/HG/H, we could define the inner product of two sections of ff and f′f' of EE to be

Adjoint of induced bundle construction

The induced bundle construction described above is a functor that takes representations of the stabilizer subgroupHH to GG-equivariant vector bundles over MM:

L:Rep(H)→Vect(M,G)L: Rep(H) \to Vect(M,G)

There is a related functor going the other way:

R:Vect(M,G)→Rep(H)R: Vect(M,G) \to Rep(H)

which restricts the action of GG on the whole bundle to the action of the stabilizer subgroup HH on the fiber over the chosen point xx. The existence of this adjunction is known as Frobenius reciprocity.

In the diagram above, on the top left we have a generic GG-equivariant vector bundle over MM, F∈Vect(M,G)F\in Vect(M,G), with projection π1:F→M\pi_1:F\to M, and a chosen point x∈Mx\in M whose stabilizer subgroup is HH. The functor RR maps FF to a representation of HH on the fiber over xx, π1−1(x)\pi_1^{-1}(x), shown on the top right.

On the bottom right, we have a generic representation of HH on a vector space VV. The morphisms of Rep(H)Rep(H) are intertwiners, so we are interested in intertwiners such as i:V→π1−1(x)i:V\to \pi_1^{-1}(x). The functor LL, the induced bundle construction, maps a generic representation of HH to a GG-equivariant vector bundle (G×V)/H(G\times V)/H, shown on the bottom left. This bundle has a projection π2:(G×V)/H→G/H\pi_2: (G\times V)/H \to G/H, π2([(g,v)])=gH\pi_2([(g,v)])=g H. Since M≅G/HM \cong G/H, this bundle is in Vect(M,G)Vect(M,G). And we are interested in the morphisms of Vect(M,G)Vect(M,G), such as (f,m)(f,m) where f:L(V)→Ff:L(V)\to F and m:G/H→Mm:G/H\to M.

In fact, we need to work with a subcategory of Vect(M,G)Vect(M,G) in which all morphisms preserve the point x∈Mx\in M. When we deal with bundles over G/H≅MG/H \cong M, we will use the obvious bijection gH→g⋅xg H \to g\cdot x, and accordingly restrict ourselves to vector bundle morphisms that map xx to the coset eHe H or vice versa.

We are assuming that GG acts transitively on MM, so given any y∈My\in M there exists at least one element of GG, say k(y)k(y), such that k(y)⋅x=yk(y)\cdot x = y. We will now assume that some definite function k:M→Gk:M\to G has been chosen with this property, and for convenience we will further assume that k(x)=ek(x)=e, the identity element in GG. The group element k(y)k(y) gives us a specific way to use the action of GG on MM to get from our chosen point xx to some other point yy — and equally, to use the action of GG on the whole bundle FF to get from the fiber over xx to the fiber over yy.

Now, to show that LL and RR are adjoint functors, we need to construct a bijection between the intertwiners i:V→π1−1(x)i:V\to \pi_1^{-1}(x) and the GG-equivariant vector bundle morphisms (f,m)(f,m), where f:L(V)→Ff:L(V)\to F and m:G/H→Mm:G/H\to M.

Given an intertwiner i:V→π1−1(x)i:V\to \pi_1^{-1}(x), we start by defining m:G/H→Mm:G/H\to M by:

m(gH)=g⋅xm(g H)=g\cdot x

which is independent of ii, and is just the obvious bijection between G/HG/H and MM. Next, we define f:L(V)→Ff:L(V)\to F by:

f([(g,v)])=g⋅i(v)f([(g,v)]) = g\cdot i(v)

In other words, given the equivalence class [(g,v)][(g,v)] we use the intertwiner ii to take v∈Vv\in V to π1−1(x)\pi_1^{-1}(x), and then the action of GG on FF to take the result to the fiber π1−1(g⋅x)\pi_1^{-1}(g\cdot x). This satisfies the compatibility condition on the projections:

We can also demonstrate a bijection between intertwiners and GG-equivariant vector bundle morphisms in the other direction: intertwiners i*:π1−1(x)→Vi^*:\pi_1^{-1}(x)\to V and vector bundle morphisms (f*,m*)(f^*,m^*), where f*:F→L(V)f^*:F\to L(V) and m*:M→G/Hm^*:M\to G/H.

Given an intertwiner i*:π1−1(x)→Vi^*:\pi_1^{-1}(x)\to V, we define m*:M→G/Hm^*:M\to G/H as:

for each w∈Fw\in F. Because k(π1(w))⋅x=π1(w)k(\pi_1(w))\cdot x = \pi_1(w), k(π1(w))−1k(\pi_1(w))^{-1} will map the entire fiber to which ww belongs to π1−1(x)\pi_1^{-1}(x), the domain of the intertwiner i*i^*. And we have:

The map f*f^* is a linear map between the fibers π1−1(y)\pi_1^{-1}(y) and π2−1(m*(y))\pi_2^{-1}(m^*(y)), because, along with the linearity of i*i^*, the vector space structure on the fibers of L(V)L(V) is defined so all maps of the form v→[(g,v)]v\to [(g,v)] are linear. So, m*m^* and f*f^* together give us a vector bundle morphism from FF to L(V)L(V).

In order to be a morphism in the category of GG-equivariant vector bundles, f*f^* should also commute with the action of GG. We have:

We make use of the linear bijection ϕe:V→EeH\phi_e:V\to E_{e H}, defined by ϕe(v)=[(e,v)]\phi_e(v)=[(e,v)]. We introduced these linear bijections ϕg\phi_g when initially describing the induced bundle construction. We define i*:π1−1(x)→Vi^*:\pi_1^{-1}(x)\to V by:

i*(w)=ϕe−1(f*(w))i^*(w) = \phi_e^{-1}(f^*(w))

We check that this is an intertwiner between the relevant representations of HH:

Examples and Applications

Regular representation

The regular representation of a group GG, as a linear representation, is the induced representation of the trivial representation along the trivial subgroup inclusion Ind1G(1)Ind_1^G(1).

Centralizer algebra / Hecke algebra

Let

i:H↪G
i \colon H \hookrightarrow G

be a group homomorphism (often assumed to be a subgroup inclusion, and sometimes with GG assumed to be a finite group). For E∈HRepE \in H Rep some HH-representation (often taken to be the trivial HH-representation), let IndiE∈GRepInd_i E \in G Rep be the induced GG-representation. Then the endomorphism ringEndG(IndiE)End_G(Ind_i E) of IndiEInd_i E in GRepG Rep is called the centralizer algebra or also the Hecke algebra or Iwahori–Hecke algebraHecke(E,i)Hecke(E,i) of the induced representation. (Basics are in (Woit, def. 2), details are in (Curtis-Reiner, section 67), a quick survey of related theory is in (Srinivasan)).

For V∈Act(G)V \in Act(G) any other representation, there is a canonical ∞-action of Hecke(E,i)Hecke(E,i) on ∏BG[∑BiE,V]\underset{\mathbf{B}G}{\prod} \left[\underset{\mathbf{B}i}{\sum} E , V \right]. If here EE is the trivial representation then by adjointness this is the invariantsVGV^G of VV and hence the Hecke algebra acts on the invariants. (See for instance (Woit, def. 2)). This is sometimes called the Hecke algebra action on the Iwahori fixed vectors (e.g. here, p. 9)